# How to tell if something has a common factor

1. Feb 4, 2010

### rsala004

if we have 6 and 9

and we break them down to sets of prime factors {2,3} , and {3}

if the intersection of the 2 sets is empty..does this mean that numbers have no common factors?

or in more specific to my interest...

if we have P and Q and we have their sets of prime factors, if the intersection is empty does this mean P/Q is an irreducible fraction?

or is this only true for their sets of non-zero prime+composite factors?

thanks

2. Feb 4, 2010

### mathman

I don't know what you mean by

3. Feb 4, 2010

### Mensanator

By definition, a set of prime factors won't contain any composites.

4. Feb 4, 2010

### CRGreathouse

I'm going to guess the question and then answer what I thought the question was.

Question: Given positive integers m and n, if the set M of primes dividing m and the set N of primes dividing n are disjoint (have an empty intersection), is m/n an irreducible fraction? [If this is false, is it at least the case that if the set M' of divisors of m and the set N' of divisors of n have {1} as their intersection, is m/n an irreducible fraction?]

Answer: Yes, if there are no primes in common than m/n is irreducible.

5. Feb 5, 2010